A) Prove that the sum of a specified element of one set is greater than
or equal to a specific number (n + 1)/2; B) Find all the ordered pairs of
integers (m, n) that satisfy the equation (n^3 + 1) / (mn - 1).

A 6-square by 6-square board is tiled completely with 18 2x1 dominoes.
Prove that at least one horizontal or vertical line can be drawn along
the edges of the dominoes that divides the board into 2 regions, without
cutting any dominoes in half.

At a country club 35 people play golf, 28 swim, and 24 play tennis. Of
these, 6 play golf and tennis only, 9 play golf and swim only, and 7 play
tennis and swim only. 8 people do all three. How many members are there
altogether?

Prove: Assume that all points in the real plane are colored white or
black at random. No matter how the plane is colored (even all white or
all black) there is always at least one triangle whose vertices and
center of gravity (all 4 points) are of the SAME color.

If multiple small equilateral triangles are drawn within a larger one,
what is the relation between the number of small triangles lying on the
base of the big triangle and the total number contained within the big
triangle?